In e na ional Jou nal o Ad anced Scien i ic and Technical Resea ch ISSN 2249-9954
A ailable online on h p://www. spublica ion.com/ijs /index.h ml olume 15, No. 5, 2025
DOI: 10.5281/zenodo.17314701
O iginal A icle
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A Scale- ela i e App oach o Uni ica ion: Analogical Equi alence o
Fundamen al Fo ces om A omic o Cosmic Scales
Anki Kuma Singh
(Co esponding Au ho ) New Delhi
ARTICLE INFO
ABSTRACT
©2025 RS Publica ion
Pape ID: IJASTR-
68E4ECF39A3EC
Recei ed: 2025-09-10
Published: 2025-10-10
DOI:
h ps://dx.doi.o g
/10.5281/zenodo.17
314701
Page No: 318-322
This pape p oposes a heo e ical amewo k sugges ing ha he ou
undamen al o ces o na u e, g a i a ional, elec omagne ic, s ong
nuclea , and weak nuclea a e in e ela ed h ough a p inciple o scale
ela i i y. D awing an analogy be ween he s uc u e o a oms and he sola
sys em, he s udy examines whe he mac oscopic g a i a ional and
magne ic in e ac ions a e scaled analogues o mic oscopic nuclea and
elec omagne ic in e ac ions. By explo ing Keple ’s laws o plane a y
mo ion as a mac o-scale e lec ion o a omic beha io , a co espondence
model is p oposed ha maps o ce ypes ac oss physical scales. The
appa en sepa a ion o undamen al o ces is hypo hesized o be an a i ac
o obse a ional scale a he han a ue di ision in na u e. The model is
o malized ma hema ically using scaling unc ions and dimensional
analysis, and illus a ed h ough diag ams. Implica ions include he
p ospec o a scale-dependen g and uni ica ion, p edic ions o
as ophysical esonances, and es s based on dimensionless in a ian s.
Keywo ds: Scale Rela i i y, Uni ied Field Theo y, Keple ’s Law, A omic–
Cosmic Analogy, Fundamen al Fo ces, G a i y–Weak Equi alence, Mass–
Cha ge Duali y
Ci e This Pape : Anki Kuma Singh (2025). "Scale-Rela i i y based uni ica ion o he
ou Fundamen al Fo ces". INTERNATIONAL JOURNAL OF ADVANCED SCIENTIFIC
AND TECHNICAL RESEARCH (IJASTR), ol. 15, no. 5, 2025, pp. 318-322. DOI:
h ps://dx.doi.o g/10.5281/zenodo.17314701
In e na ional Jou nal o Ad anced Scien i ic and Technical Resea ch ISSN 2249-9954
A ailable online on h p://www. spublica ion.com/ijs /index.h ml olume 15, No. 5, 2025
DOI: 10.5281/zenodo.17314701
O iginal A icle
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In oduc ion
The sea ch o a uni ied desc ip ion o na u e’s undamen al o ces is cen al in heo e ical
physics. G a i y, elec omagne ism, he s ong nuclea , and he weak nuclea o ce ha e
adi ionally been ea ed as dis inc , bu ecu ing s uc u es on a omic o galac ic scales
sugges he possibili y o unde lying symme ies epea ed ac oss magni udes o scale. This
s udy in oduces a scale-based equi alence hypo hesis, posi ing ha hese o ces a e
di e en mani es a ions o a single in e ac ion, modula ed by obse a ional scale. The
analogy be ween a omic and cosmic s uc u es, especially in dynamics and con igu a ion,
o ms he a gumen ’s basis.
Concep ual F amewo k
A om–Sola Analogy
Jus as a c icke s adium models he scale o an a om, wi h he nucleus as a ennis ball a he
pi ch and elec ons nea he ou ield, so oo does he sola sys em, wi h i s massi e, sun-like
“nucleus” and plane a y “elec ons”, e lec his spa ial and dynamical s uc u e. The s abili y
in bo h sys ems a ises om a balance o cen ipe al and a ac i e o ces, and ellip ical o bi s
(plane a y and a omic) a e go e ned by analogous laws: Keple 's laws and he p obabilis ic
shapes o a omic o bi als.
Scale Rela i i y P inciple
The scale ela i i y p inciple asse s ha he o ms o physical laws a e p ese ed, once
app op ia e scale ac o s a e applied, ac oss a omic, plane a y, and galac ic domains. The
cons an s ha domina e each egime (such as ℏ and G) shi wi h scale, bu he unde lying
dynamics a e s uc u ally simila . This leads o he hypo hesis ha each o ce, a each scale,
is he scale- ela i is ic image o ano he .
Fo ce Equi alence
Mac ocosmic Fo ce , Mic ocosmic Analogue , P ima y Cha ac e is ic , P oposed
Equi alence
G a i a ional Fo ce , Weak Nuclea Fo ce , Long- ange a ac ion ia mass-ene gy cu e ,
In e na ional Jou nal o Ad anced Scien i ic and Technical Resea ch ISSN 2249-9954
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DOI: 10.5281/zenodo.17314701
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Fg ∼ Fw
Elec omagne ic Fo ce , S ong Nuclea Fo ce , Binding ia cha ge/magne ic e ec s , Fe ∼
Fs
When mapped h ough app op ia e scaling ans o ma ions, g a i y mi o s he weak o ce,
and elec omagne ism o de s as he s ong o ce a educed (nuclea ) scales. These
co espondences a e p oposed as he key symme y unde lying appa en o ce di e si y.
Ma hema ical Modeling
Dimensional Se up
Le scale a ios compa e mac o (capi alized) and mic o (lowe case) quan i ies:
SL = Lmac o / Lmic o, SM = Mmac o / Mmic o, ST = Tmac o / Tmic o
Fo ce dimension: [MLT−2]; he e o e, he scaling ac o o o ce is:
SF = SM SL ST−2
Empi ical Scale Es ima es
Boh adius: a0 = 5.29 × 10−11 m
Ea h's o bi al adius: RE = 1.49 × 10^11 m
SL = RE / a0 ≈ 2.8 × 10^21
P o on adius: p = 0.84 × 10−15 m
Sun adius: R⊙ = 6.96 × 10^8 m
SL(Sun/p o on) = 8.3 × 10^23, SM(Sun/p o on) = 1.19 × 10^57
Scaling Func ion
De ine he mapping: φ(S) = SM^p SL^(1−p), 0 ≤ p ≤ 1
Fmac o = φ(S) Fmic o
This unc ion adjus s undamen al cons an s o scale, e.g.:
G′ = φ(S)G, ℏ′ = ℏ / φ(S)
Mapping Be ween Fo ces
Mass–Cha ge Analogy
Le e ec i e cha ge qe = βm, gi ing a Coulomb-like o ce:
FCe = (1 / 4πε0) * β² * m1m2 / ²
Rela e o New onian g a i y:
Gm1m2 / ² = (β² / 4πε0) * m1m2 / ² ⇒ β = √(4πε0G) ≈ 8.62 × 10−11 C/kg
Thus, g a i y a one scale can be in e p e ed as a scaled image o elec omagne ic
in e ac ion.
In e na ional Jou nal o Ad anced Scien i ic and Technical Resea ch ISSN 2249-9954
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DOI: 10.5281/zenodo.17314701
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Weak–G a i y Scaling
I he weak o ce sa is ies Fw ∝ Gm² / ², hen mac o g a i y and weak mic o-in e ac ions
a e scale-linked h ough φ(S).
Keple -Boh Time Scaling
Keple ’s 3 d law o o bi al pe iod:
T² = 4π²a³ / GM
Boh ’s o bi al pe iod:
TBoh = (2πℏ³) / (Z e² m e²)
Rela ing pe iods by scaling:
ST ∼ SL^(3/2) SM^(−1/2) (Gmac o / Gmic o)^(1/2)
This supplies he ime scaling in φ(S).
Implica ions and Tes s
Uni ica ion: Sugges s a scale-dependen G and Uni ied Theo y (SD-GUT) pa hway.
Quan um G a i y: Poin s o g a i a ional quan iza ion h ough in e se weak o ce scaling.
As ophysical P edic ions: Magne ic o g a i o-magne ic esonances in plane a y sys ems
may mi o a omic ene gy shells.
Dimensionless In a ian s: I speci ic o ce a ios a e p ese ed ac oss scale, his con i ms
sel -simila i y.
Mass–Cha ge P opo ionali y: Compa e q/m a ios o cosmic plasmas o p oximi y o β.
Limi a ions and Fu u e Wo k
The exac o m o he scaling unc ion φ(S) mus be igo ously de i ed om
eno maliza ion heo y.
Quan um and ela i is ic e ec s a e beyond he p esen analysis.
La ge-scale simula ions a e necessa y o alida e scale-induced quan iza ion and
pe iodici y in as ophysical se ings.
Nex S eps:
- Cons uc eno maliza ion-g oup (RG) models o coupling lows.
In e na ional Jou nal o Ad anced Scien i ic and Technical Resea ch ISSN 2249-9954
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DOI: 10.5281/zenodo.17314701
O iginal A icle
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- Simula e N-body g a i a ional sys ems wi h quan iza ion imposed.
- Analyze exoplane a y da a o o bi al pe iod clus e ing.
Conclusion
A scale- ela i i y app oach is in oduced linking all undamen al o ces h ough analogical
and ma hema ical equi alence, sugges ing ha wha appea as independen in e ac ions a e,
in ac , scale- ans o med ace s o a single law. Pa allels be ween a omic and cosmic
s uc u es p o ide a ounda ion o his hypo hesis, opening a enues o scale-uni ying
o malism and simula ion-based e i ica ion.
